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Abstract

Fluid flow chronologically is widely recognized due to its various uses in turbines, the framework of spinning magnet stars, gyromagnetic generators, and chemical engineers observing the progression of petroleum through the aquifer, and blood vessels in the respiratory alveolar plate. Tropical cyclones, pools of water, and storms all exhibit rotational movement. The current investigation aims to analyse the micro polar fluid flow between two infinite vertical discs enclosing Hall impact, varying thermal conductivity, heat flux as well as anomalous heat generation. The implication of a chemical change combined with chemical potential improves mass propagation. Suitable similarity conversions are used to convert the defined problems into conventional differential equations (ODEs). Furthermore, by introducing new variables the ODEs are transformed into nonlinear coupled ODEs and then solved numerically by the RK 4^th order along with the shooting technique. The velocity profiles decrease as suction parameter increases. The temperature field exhibits a rising behaviour for the increasing values of thermophoresis, Brownian and radiations parameters while the concentration field shows a decreasing behaviour. Shear stress at the upper wall increases when the rotation variable and suction variable are augmented. Heat transmission escalations at the bottom wall when Prandtl number and radiation factor are enhanced. The novelty of the present work is to examine the Buongionro model in the presence of a heat source and chemical reaction inside the Darcian porous rotating channel, which has not been investigated yet. In some limiting cases, a comparison of the on-going study with existing literature is also included to justify the contemplated problem.

Keywords

variable magnetic field thermal radiation heat source temperature conductivity nanofluid porous stretching sheet

Introduction

These days, researchers are very interested in applications of nanofluid and heat transfer enhancement techniques. According to empirical findings, the substantial properties of heat flow, mass transfer, and density during the flow are present in working fluids in a variety of technical and medical disciplines. Although different base liquids can transfer heat with a similar amount of efficiency, these liquids are often not ideal for thermal transport applications because of their low thermal ability. This problem can be resolved by using certain nanoparticle additions to increase the heat efficiency of such conventional approaches. The most effective way to improve heat transportation, which is employed in a variety of engineering products and air-conditioning systems, is to suspend ordinary fluids with nanosize particles. This idea allows for the replacement of convectional fluids with lower heat efficiency with nanofluids that can transfer more heat to surfaces. Nano fluids are typically more stable in terms of heat and momentum transmission than regular fluids. Different metals, including nitrides, oxides, borides, gold, steel, silver, and copper, can be used cautiously to improve such nanomaterials.

The study of boundary-layer flow towards a stretching surface was first introduced by Anderson¹ has become a significant and exciting subject for research investigations because of its wide industrial application such as cooling capacity polymer, heat exchange properties such as thermal performance, wire coating-layer, liquids diffusivity, and chemical production sectors. Sakiadis^2,3 examined the boundary layer flow over flat sheet for laminar and turbulent flow. Further, Crane⁴ expanded the problem to analyse the varying extending sheet velocity and closed form analytically solutions was obtained for Navier–Stokes system using two dimensional flow. These initial assistances have been of excessive significance in wire performance and elastic fabrication industries. The temperature dissemination of the melting fluid has been deliberated,⁵ where the outcomes are reflected very imperative in the development of conserving polymer sheets extracted through the die. Wang⁶ amended the fluid dynamic problem outside the stretched tube, which is required for the manufacturing environment of wire and fibers pulling. Ishak⁷ recently expanded the study of Wang to investigating melting heat rate over a permeable surface. Keeping the applications of the stretching sheet, many researchers analysed the fluid flow and heat transfer over the stretching sheet under various conditions. For this goal, the nanoparticles were introduced in the base fluid to improve the heat transfer characteristics such as viscosity, thermal efficiency, and diffusivity in the fluid.

It is witnessed that chemical processes have significant influence on heat transfer in various branches of science and nanotechnology like solar collectors, nuclear reactor safety, metallurgy and chemical engineering.^8–10 Nadeem et al.¹¹ considered the consequence of chemical performance for blood flow consuming Walter's B model over a tapering artery. Hayat et al.¹² scrutinized the chemical processes for Maxwell fluid. Using the BLF, Abdul et al.¹³ inspected the chemical processes over stretchable sheet filled with nanofluid. The homogeneous and heterogeneous chemical processes over the permeable stretching surface were studied by Kameswaran et al.¹⁴ Uddin et al.¹⁵ numerical investigated the MHD convective BLF with slip effect over stretching porous plate with chemical processes. The analytical solution was obtained for free convective flow (FCF) over a horizontal surface filled with nanofluid.¹⁶ Most recently, Srinivas et al.¹⁷ considered the viscoelastic fluid over stretching pipe with chemical processes. Similarly, the MHD FCBLF with chemical reaction of NF through a vertical surface was explored by Uddin et al.¹⁸ Patil et al.¹⁹ investigated the magnetized chemically hybrid nanofluid with thermal over stretched with Ohmic heating and dissipation effect. Shamshuddin et al.²⁰ studied the thermal exploration of ethylene glycol nanofluid movement with viscosity effect over a permeable cylindrical annulus. Shamshuddin et al.²¹ explored the thermo-solutal and chemical reaction impacts on magnetized nanofluid movement along a vertical stretching sensor surface. Patil et al.²² examined the Powell Eyring nanoliquid movement over a porous elongated surface with inclined magnetic field and chemical reaction.

In the present study, we investigate the nanofluid flow over the parallel plate horizontally in the presence of chemical reaction, Hall current, and linear radiant energy. The characteristics of fluid dynamic is also examined with variable thermal and heat generation and absorption. The model partial differential equations are altered to ordinary differential equation by similarity variables. The ODEs are transformed into first order ODEs through new variables and then solve numerically. The novelty of the present work is that no one analyses the influence of chemical reaction using the ferrofluid in two parallel horizontal plates in the presence of hall current. So, in limiting case, the present work is validated to the published work. The influence of involving parameters is explained through graphs and tables on the flow characteristics. For the validation the present work is compared to the published work reported by Dutt et al.⁵ and good analogous is found.

Mathematical formulation

Incompressible nanofluid is examined over the two parallel porous horizontal plates in the presence of chemical reaction and hall current. The plate is spinning with the velocity $Ω = (0, Ω_{1}, 0)$ towards the $y - a x i s$ . Thermal radiation is also considered in the fluid flow dynamics. The equations of velocity, temperature and concentration are modeled in the Cartesian coordinates system in which $x - a x i s$ is parallel to the plate, the $y - a x i s$ is taking tangential, and the $z - a x$ is slanting to the $x y -$ plane as shown in Figure 1. The fluid is electrically conducting due to the presence of magnetic field. The lower plate is moving with velocity $u_{w} = c x$ towards the x-direction at $y = 0$ , and the upper plate is at rest at $y = h$ . Noted that at $v = - v_{0} (v_{0} > 0$ for suction and $v_{0} < 0$ is for injection).

Figure 1.

Flow geometry of the model.

Figure 2.

Flow chart for the numerical scheme.

The basic flow equations of nanofluid are^2,3,18:

\frac{\partial \tilde{u}}{\partial x} + \frac{\partial \tilde{v}}{\partial y} = 0

(1)

\tilde{u} \frac{\partial \tilde{u}}{\partial x} + \tilde{v} \frac{\partial \tilde{u}}{\partial y} + 2 Ω_{1} \tilde{w} = - \frac{1}{ρ} \frac{\partial ρ}{\partial x} + ν (\frac{\partial^{2} \tilde{u}}{\partial x^{2}} + \frac{\partial^{2} \tilde{u}}{\partial y^{2}}) + \frac{σ_{1} B_{0}^{2}}{ρ (1 + m^{2)}} (\tilde{u} - m \tilde{w}),

(2)

\tilde{u} \frac{\partial \tilde{v}}{\partial x} + \tilde{v} \frac{\partial \tilde{v}}{\partial y} = - \frac{1}{ρ} \frac{\partial ρ}{\partial y} + ν (\frac{\partial^{2} \tilde{v}}{\partial x^{2}} + \frac{\partial^{2} \tilde{v}}{\partial y^{2}}),

(3)

\tilde{u} \frac{\partial \tilde{w}}{\partial x} + \tilde{v} \frac{\partial \tilde{w}}{\partial y} - 2 Ω_{1} \tilde{u} = ν (\frac{\partial^{2} \tilde{w}}{\partial x^{2}} + \frac{\partial^{2} \tilde{w}}{\partial y^{2}}) - \frac{σ_{1} B_{0}^{2}}{ρ (1 + m^{2)}} (m \tilde{u} + \tilde{w}),

(4)

\begin{aligned} \tilde{u} \frac{\partial \tilde{T}}{\partial x} + \tilde{v} \frac{\partial \tilde{T}}{\partial y} = \frac{1}{ρ C_{P}} \frac{\partial}{\partial y} [k (T) \frac{\partial \tilde{T}}{\partial y}] + τ [D_{B} \frac{\partial \tilde{C}}{\partial y} \frac{\partial \tilde{T}}{\partial y} + \frac{D_{T}}{{\tilde{T}}_{l}} {(\frac{\partial \tilde{T}}{\partial y})}^{2}] \\ - \frac{1}{ρ c_{p}} \frac{\partial q_{r}}{\partial y} + \frac{K (T) u_{w}}{x v ρ c_{p}} [D ({\tilde{T}}_{w} - {\tilde{T}}_{l}) f^{'} + H (\tilde{T} - {\tilde{T}}_{l})], \end{aligned}

( 5)

\tilde{u} \frac{\partial \tilde{C}}{\partial x} + \tilde{v} \frac{\partial \tilde{C}}{\partial y} = D_{B} [(\frac{\partial^{2} \tilde{C}}{\partial x^{2}} + \frac{\partial^{2} \tilde{C}}{\partial y^{2}}] + \frac{D_{T}}{{\tilde{T}}_{l}} (\frac{\partial^{2} \tilde{T}}{\partial x^{2}} + \frac{\partial^{2} \tilde{T}}{\partial y^{2}}) - k_{r}^{2} (\tilde{C} - {\tilde{C}}_{l}) (\frac{\tilde{T}}{{\tilde{T}}_{l}})^{n} \exp (\frac{- E_{a}}{k \tilde{T}})

(6)

With boundary conditions^4,5,18

\tilde{u} |_{y = 0} = u_{w} = cx, \tilde{v} |_{y = 0} = - v_{0}, \tilde{w} |_{y = 0} = 0, \tilde{T} |_{y = 0} = {\tilde{T}}_{w}, \tilde{C} |_{y = 0} = {\tilde{C}}_{w}

(7)

\tilde{u} |_{y = h} = 0, \tilde{v} |_{y = h} = 0, \tilde{w} |_{y = h} = 0, \tilde{T} |_{y = h} = {\tilde{T}}_{l}, \tilde{C} |_{y = h} = {\tilde{C}}_{l} .

The radioactive heat flux is expressed by the following relation^3,4

q_{r} = - \frac{4}{3} \frac{\bar{}}{\bar{k}} \frac{\partial T^{4}}{\partial y}, w h e r e T^{4} = 4 T_{l}^{3} T - 3 T_{l}^{4}

(8)

The varying thermal conductivity given in equation (5) is explored as^4,8:

K = k_{0} (1 + d (\frac{T - T_{l}}{T_{w} - T_{l}}))

(9)

By using equations (8) and (9), equation (5) becomes

\begin{aligned} \tilde{u} \frac{\partial \tilde{T}}{\partial x} + \tilde{v} \frac{\partial \tilde{T}}{\partial y} = \frac{1}{ρ C_{P}} \frac{\partial}{\partial y} [k_{0} (1 + d θ) \frac{\partial \tilde{T}}{\partial y}] + τ [D_{B} \frac{\partial \tilde{C}}{\partial y} \frac{\partial \tilde{T}}{\partial y} + \frac{D_{T}}{{\tilde{T}}_{l}} {(\frac{\partial \tilde{T}}{\partial y})}^{2}] \\ + \frac{16}{3} \frac{\bar{}}{\bar{k}} \frac{T_{l}^{3}}{ρ C_{P}} T \frac{\partial^{2} T}{\partial y^{2}} + \frac{k_{0} (1 + d θ) u_{w}}{x v ρ c_{ρ}} [D ({\tilde{T}}_{w} - {\tilde{T}}_{l}) f^{'} + H (\tilde{T} - {\tilde{T}}_{l})], \end{aligned}

(10)

By proper conversion^2,3

\tilde{u} = c x f^{'} (η), \tilde{v} = - c h f (η), \tilde{w} = c x j (η), θ (η) = \frac{\tilde{T} - {\tilde{T}}_{l}}{{\tilde{T}}_{w} - {\tilde{T}}_{l}}, (η) = \frac{\tilde{C} - {\tilde{C}}_{l}}{{\tilde{C}}_{w} - {\tilde{C}}_{l}}, η = \frac{Y}{H} .

(11)

By using Eq. (11) the Eq. (1) satisfy automatically and Eqs. (2-4, 6, 7, and 10) yield the system

\frac{d^{4} f}{d η^{4}} = R_{e} (\frac{d f}{d η} \frac{d^{2} f}{d η^{2}} - f \frac{d^{3} f}{d η^{3}}) + 2 α_{1} \frac{d j}{d η} - \frac{H a^{2}}{(1 + m^{2})} (\frac{d^{2} f}{d η^{2}} - m \frac{d j}{d η}),

(12)

\frac{d^{2} j}{d η^{2}} = R_{e} (f \frac{d j}{d η} - f \frac{d f}{d η} j) - 2 α_{1} \frac{d f}{d η} + \frac{H a^{2}}{(1 + m^{2})} (j + m \frac{d f}{d η}),

(13)

((1 + d θ) + \frac{4}{3} R d) R_{e}) \frac{d^{2} θ}{d η^{2}} = - P r (f \frac{d θ}{d η} + R_{e} (N b \frac{d θ}{d η} \frac{d ϕ}{d η} + N t {(\frac{d θ}{d η})}^{2})) - (1 + d θ) (D \frac{d f}{d η} + H \frac{d θ}{d η}) - R_{e} d {(\frac{d θ}{d η})}^{2}

(14)

\frac{d^{2}}{d η^{2}} = - \frac{N t}{N b} \frac{d^{2} θ}{d η^{2}} + Sc (δ ϕ (1 + α θ)^{n} \exp (\frac{- E}{(1 + α θ)}) - f . R_{e} \frac{d ϕ}{d η}) .

(15)

The boundary constraints take the form.

Lower plate:

\frac{d f}{d η} (0) = 1, f (0) = K, j (0) = 0, θ (0) = 1, ϕ (0) = 1,

Upper plate:

\frac{d f}{d η} (1) = 0, f (1) = 0, j (1) = 0, θ (1) = 0, ϕ (1) = 0.

(16)

The physical quantities of interest are as follow;

C_{f, l o w e r} = \frac{μ}{ρ u_{w}^{2}} \frac{\partial u}{\partial y} |_{y = 0}

(17)

C_{f, u p p e r} = \frac{μ}{ρ u_{w}^{2}} \frac{\partial u}{\partial y} |_{y = h}

(18)

N u_{l o w e r} = \frac{h Q_{w}}{k_{0} (T_{w} - T_{l})}, Q_{w} = - k (T) \frac{\partial T}{\partial y} + q r |_{y = 0}

(19)

N u_{u p p e r} = \frac{h Q_{w}}{k_{0} (T_{w} - T_{l})}, Q_{w} = - k (T) \frac{\partial T}{\partial y} + q r |_{y = h}

(20)

S h_{l o w e r} = \frac{h Q_{m}}{D_{B (C_{w} - C_{l})}}, Q_{m} = - D_{B} \frac{\partial C}{\partial y} |_{y = 0}

(21)

S h_{l o w e r} = \frac{h Q_{m}}{D_{B (C_{w} - C_{l})}}, Q_{m} = - D_{B} \frac{\partial C}{\partial y} |_{y = h}

(22)

Using Eq. (11) in Eqs. (17-22) we get:

(R e_{h} C_{f})_{f l o w e r} = \frac{d^{2} f}{d η^{2}} |_{η = 0}, (R e_{h} C_{f})_{u p p e r} = \frac{d^{2} f}{d η^{2}} |_{η = 1}

(23)

(N u)_{l o w e r} = - (1 + (\frac{4}{3}) . \frac{R d}{1 + d θ}) \frac{d θ}{d η} |_{η = 0}, (N u)_{u p p e r} = - (1 + (\frac{4}{3}) . \frac{R d}{1 + d θ}) \frac{d θ}{d η} |_{η = 1}

(24)

(S h)_{l o w e r} = - \frac{d φ}{d η} |_{η = 0}, (S h)_{u p p e r} = - \frac{d φ}{d η} |_{η = 1}

(25)

Numerical procedure

Due to higher nonlinear equations of velocity, temperature and concentration profiles, the exact solution is impossible. For this purpose the numerical solution has been obtained for the flow characteristics. For the PDEs are converted to ODEs by transformation variables. ODEs are then converted to first order ODEs through new variables and then solved numerically by applying RK4 method wit step size $Δ η = 0.001$ and convergence to 10⁻⁵. Figure is depicted the flow diagram of RK4 scheme.

Results and discussion

In this section we will investigate the influence of physical parameters on velocity, temperature, and concentration profiles graphically and tabular form. Equations (12)-(15) show visually the characteristics of the velocities $f^{'} (η), j (η),$ temperature $θ (η)$ , and concentration $ϕ (η)$ for the dimensionless factors occurring in the severely complex scientific and analytical problem. With the help of the MATLAB program RK4, this design is numerically evaluated. The fluid becomes greater pressurized as a consequence. The impact of adjusting the Suction factor K on velocities is seen in Figures 3(a) and 3(b). A significant volume of fluid is discharged in the vicinity of a lower plate when it suctions nanoliquid. As a result, a sense of rising K is interpreted as falling $f^{'} (η)$ and $j (η)$ . Figures 4(a) and 4(b) show how the rotation factor affects the variables $f^{'} (η), j (η),$ temperature $θ (η)$ , and concentration $ϕ (η)$ .

Figure 3.

(a) Influence of K on $f^{'} (η)$ . (b) Influence of K on $J (η)$ .

Figure 4.

(a) Influence of a on $f^{'} (η)$ . (b) Influence of a on $J (η)$ .

The movement of the liquid is resisted by the Coriolis force, which acts in a revolving channel transverse to the velocity profiles and the spinning axis. There is a dual impression for $f^{'} (η)$ as a result. When is reinforced, the velocity $f^{'} (η)$ decreases in the closest area of the lower plane while rising in the network's highest region. It is seen that the velocity $j (η)$ drops as grows. The effect of the magnetic factor Ha on $f^{'} (η)$ and $j (η)$ is seen in Figures 5(a)–5(b). As Ha values rise, the Lorentz force, which resists fluid flow, rises as well. A drop can be seen in Fig. 5(a) before the network's midway, but here is a rising tendency in the channel's upper half. As can be seen in Figure 5(b), when Ha is changed, the fluid movement begins to degrade.

Figure 5.

(a) Influence of $H a$ on $f^{'} (η)$ . (b) Influence of $H a$ on $J (η)$ .

The Reynolds number $R e$ impression is seen in Figures 6(a)–6(b). Since the ratio of centrifugal to viscous force is represented by the Reynolds number. Inertial forces thus increase as Re increases, but viscous forces decrease. For $f^{'} (η)$ , is proven to have a dual affect, yet velocity $j (η)$ declines. The behaviour of the Hall current factor m on $f^{'} (η)$ and $j (η)$ is shown in Figures 7(a)–7(b). When m is amplified, the functional conductivity $\frac{σ 1}{1 + m^{2}}$ significantly decreases. The impact of the magnetic retarded force is thereby much reduced. As can be observed in Fig. 7(a), the upper half of the channel exhibits an improving and maintaining order, whereas the bottom half displays a decimating tendency.

Figure 6.

(a) Influence of $R e$ on $f^{'} (η)$ . (b) Influence of $R e$ on $J (η)$ .

Figure 7.

(a) Influence of m on $f^{'} (η)$ . (b) Influence of m on $J (η)$ .

As m rises, the fluid's velocity $j (η)$ drops. The result of the thermophoresis factor Nt on () is shown in Figure 8. Nanoparticles go through high stress to low stress fluid for rising $N t$ values. It has been shown that the thermophoretic force becomes stronger with growing $N t$ . Consequently, $θ (η)$ increases. Figure 9 shows the impact N_b on $θ (η)$ . It has been shown that nanoparticle collision causes considerably swelling N_b values to generate significantly more heat. Consequently, $θ (η)$ rises. Figure 10 shows the radiation variable $R d$ abusive behaviour on $θ (η)$ . The unintentional mobility of nanomaterials is increased when $R d$ is added to the temperature. The constant collision results in the production of extra energy. There is thus an increase in $θ (η)$ . The effect of the thermal conductivity factor d on $θ (η)$ is unquestionably confirmed by Figure 11. The heat function is improved by larger d values. As a results, augmentation in $θ (η)$ is felt.

Figure 8.

(a) Impact of $N t$ on $θ (η)$ .

Figure 9.

(b) Impact of $N b$ on $θ (η)$ .

Figure 10.

(a) Impact of $R d$ on $θ (η)$ .

Figure 11.

(b) impact of d on $θ (η)$ .

Figures 12(a)-12(b) and 13(a)-13(b) demonstrate the presentation of variable heat source/ sink on $θ (η)$ . It is observed that increasing amount of the variable heat source variable $(D > 0, H > 0)$ correspond to significantly increased heat generation. As a result, temperature increases. However, when the variable sink value $(D > 0, H > 0)$ is significantly changed, the nature of $θ (η)$ deteriorates owing to inner absorption of heat. Figure 14 explores the Schmidt number $S c$ influence on the concentrated field. It is perceived that by increasing $S c$ , the concentration field decreases, owing to a lessening in mass dissemination. Figures 15 and 16 show the effect of the Brownian motion $N b$ and thermophoresis variable $N t$ on $ϕ (η)$ . As a consequence, there is an increase in $ϕ (η)$ . Figures 12(a)-12(b) and 13(a)-13(b) show how a changing heat source/sink is shown on $ϕ (η)$ . It has been shown that much more heat is produced as the variable heat source variable's value increases ( $D > 0, H > 0$ ). Temperature rises as a consequence. However, the nature of $ϕ (η)$ deteriorates due to internal heat absorption when the variable sink value $(D > 0, H > 0)$ is greatly altered. The impact of the Schmidt number $S c$ on the concentrated field is examined in Figure 14. The concentration field is thought to shrink as Sc increases due to a reduction in mass distribution. The impact of $N t$ and $N b$ on $ϕ (η)$ is seen in Figures 15 and 16. $N b$ and $N t$ drift in the reverse direction from $ϕ (η)$ . Higher values of N_t speed up the movement of fluid atoms, causing the value of $ϕ (η)$ to increase. As $N b$ rises, the stochastic mobility of the fluid particles increases. The amplifying N_b fluid concentration therefore declines. The result of the dimensionless chemical process parameter, as determined by $δ$ , is shown in Figure 17. Due to chemical consumption, overall diffusivity of chemical molecules diminishes. The fluid concentration falls as a result. The effect of producing activating energy E amounts is seen in Figure 18. It has been shown that Arrhenius function decreases with increasing E values. The impulsive chemical reaction decreases as a consequence. The concentration of the fluid rises as E does. The result of the temperature differential variable a on $ϕ (η)$ is shown in Figure 19. As the temperature variation (both near and far from the surface) increases, a worsening concentration profile is seen.

Figure 12.

(a) Impact of $D > 0$ (b) $H > 0$ on $θ (η)$ .

Figure 13.

(a) Influence of $D < 0$ (b) $H < 0$ on $θ (η)$ .

Figure 14.

(a) Influence of $S c$ on $ϕ (η)$ .

Figure 15.

(b) influence of $N b$ on $ϕ (η)$ .

Figure 16.

(a) Influence of $N t$ on $ϕ (η)$ .

Figure 17.

(b) influence of $δ$ on $ϕ (η)$ .

Figure 18.

(a) Influence of E on $ϕ (η)$ .

Figure 19.

(b) influence of a on $ϕ (η)$ .

The impact of $H a$ , $α$ , and K on shear stress at the bottom and upper walls is examined in Table 1. It is thought that as Ha grows, shear stress rises at both walls whereas, $f^{″} (0)$ and $j^{'} (0)$ fall as the value of K rises, increasing shear force at the top wall. The upper wall shearing stress rises as the rotational variable is magnified, whereas $f^{″} (0)$ has the opposite effect. Table 2 shows the impact of tabulated values for the non-dimensional factors Pr, Rd, and d on the rate of heat transfer. When $P r, R d,$ and d rise, it is seen that the heat flow also increases. Table 3 shows how different $S c, δ, E, N t$ and $N b$ values affect the rate of mass transfer. It is understood that the mass flow $ϕ^{'} (0)$ increases at the bottom wall as Sc and improve. On the other hand, the mass flow decays as $E, N t$ and $N b$ values rise. As $S c$ and $δ$ are raised, the mass transmission rate at the top wall decreases, while $E, N t$ and $N b$ respond in the opposite way to the intended result. A comparison between the present researches with Dutt et al.⁵ is shown in Table 4. The results show a strong association with present work. Figures 20 and 21 show how physical characteristics like the magnetic factor Ha, the Schmidth number (SN) $S c$ , and the Brownian motion factor $N b$ affect the Skin friction (SF) and Sherwood number. It is pretty obvious from Figure 20 that SF rises as the magnetic factor increases. This is due the Lorentz force develops when the magnetic field is applied. The SF rises as a result of Lorentz. Similar behaviour is inspected for the parameters $S c$ , and $N b$ on the SN as shown in Figure 21.

Figure 20.

(a) influence of $H a$ on skin friction. (b) Of $S c$ on Sherwood number.

Figure 21.

Impact of $N b a n d S c$ on Sherwood number.

Table 1.

Numerical values for friction drag coefficient at both wall.

Ha	$α_{1}$	$K$	$f^{″} (0)$	$J^{'} (0)$	$f^{″} (1)$	$J^{'} (1)$
0.4	1	0.2	−3.8262455	0.22541159	3.5926925	0.54849149
0.6			−3.7983194	0.22937296	3.5992994	0.58998432
0.8			−3.7619566	0.23142925	3.5987119	0.64987973
0.3	1	0.4	−3.7988799	0.28672762	3.5953999	0.97287839
	3		−3.9971633	0.37866375	3.6589899	1.5282953
	5		−4.4167316	0.43814944	3.7469865	1.9585998
0.4	1	0.1	−4.5444949	−0.1333518	3.9993977	0.67772693
		0.3	−5.9429682	−0.3575427	4.9899783	0.93912785
		0.5	−7.6632682	−0.5299198	5.9943565	1.1715158

Table 2.

Numerical values for heat transfer rate.

$P r$	$R d$	$d$	$- (1 + \frac{R d}{1 + d θ}) θ^{'} (0)$	$- (1 + \frac{R d}{1 + d θ}) θ^{'} (1)$
1	0.5	0.2	2.8582227	3.2686563
4			3.0121106	3.3056284
7			3.0652488	3.5481764
3	0.2	0.1	2.5826245	2.8735874
	0.4		2.8606061	3.2301873
	0.6		3.1282258	3.4888387
3	0.5	0.2	3.0681304	3.5778328
		0.4	3.2148877	3.0386377
		0.6	3.3585610	4.5308064

Table 3.

Numerical values for mass transfer rate.

$S c$	$N t$	$δ$	$N b$	$E$	$- ϕ^{'} (0)$	$- ϕ^{'} (1)$
0.5	0.2	0.4	0.2	0.5	1.2682748	0.8782798
1.0					1.3569924	0.8331143
1.4					1.4471383	0.7999553
1.0	0.1				1.8912389	3.7326718
	0.3				1.7123892	3.9118486
	0.5				1.5379884	4.2974655
	0.1	0.6			1.2446956	0.8882515
		1.2			1.2747579	0.8771941
		1.4			1.2944949	0.8663445
			0.2		1.9955616	3.4693693
			0.4		1.6926893	3.9924682
			0.6		1.3628933	4.5733221
				1	1.1981187	0.8961844
				2	1.1853191	0.8999332
				3	1.1791132	0.9129878

Table 4.

Validation of the present work with published work for temperature and concentration profiles.

$η$	Dutt et al.⁵ $θ (η)$	Current	Dutt et al.⁵ $ϕ (η)$	Current
0	1	1	1	1
0.1	0.7568230	0.7568230	0.6160560	0.6160560
0.3	0.6071153	0.6071151	0.3736382	0.3736381
0.5	0.4046126	0.4046123	0.1671527	0.1671527
0.7	0.1718404	0.1718401	0.0026777	0.0026778
1	0	0	0	0

Conclusions

The current investigation aims to analyse the micro polar fluid flow between two infinite vertical discs enclosing Hall impact, varying thermal conductivity, heat flux as well as anomalous heat generation. The implication of a chemical change combined with chemical potential improves mass propagation. Suitable similarity conversions are used to convert the defined problems into conventional differential equations (ODEs). Furthermore, by introducing new variables the ODEs are transformed into nonlinear coupled ODEs and then solved numerically by the RK4 approach along with the shooting technique. The novelty of the present work is to examine the Buongionro model in the presence of a heat source and chemical reaction inside the Darcian porous rotating channel, which has not been investigated yet. In some limiting cases, a comparison of the on-going study with existing literature is also included to justify the contemplated problem.

The results of the current investigation are given below:

The velocity profiles decrease as K increases.

The temperature field exhibits a rising behaviour for the increasing values of N_t, N_b and d, and Rd.

For N (b) and N t, the concentration field shows a decreasing behaviour.

The concentration field diminishes as S_c and δ are enhanced.

Shear stress at the upper wall increases when the rotation variable and suction variable are augmented.

Heat transmission escalations at the bottom wall when Pr, Rd, and d are amplified.

The current investigation can be enhanced to include Hall current as well as Ion slip effects along with any viscoelastic non-Newtonian model. The Non-Newtonian models have wide range of applications in the fluid realm. Additionally, simple thermal radiation can be replaced with nonlinear thermal radiation, and a gyrotactic microorganism's effect can be added.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Zeeshan

Data availability statement

The data presented in this study are available upon request from the corresponding authors.

Author biographies

Zeeshan is working as an Assistant Professor in Mathematics, Department of Mathematics and Statistics, Bacha Khan University Charsadda, Pakistan. He did his PhD from Abdul Wali Khan University Mardan. He received his MPhil and Master degree in mathematics from Quaid-i-Azam University Islamabad. He has published more than 80 papers in JCR journals.

Muhammad Shoaib Khan is working as an Assistant Professor in Mathematics, Department of Mathematics and Statistics, Bacha Khan University Charsadda, Pakistan. He did his PhD from Peshawar University. He received his MPhil and Master degree in mathematics from COMSATS University Islamabad and Bacha Khan University Charsadda. He has published more than 80 papers in JCR journals.

Ilyas Khan received his PhD degree in applied mathematics from the Universiti Teknologi Malaysia(UTM) in 2012 and Post Doctorate in 2013 from the same university. He received several distinctions such as senior visiting research fellow and visiting researcher. He has over 15 years of academic experience in different reputed institutions of the world. He is currently working as an Associate Professor in the Department of Mathematics, College of Science, at Majmaah University, Saudi Arabia. He has received twice distinguish researcher award from Majmaah University. According to AD-Scientific Index 2022, He has received first place as a best scientist at Majmaah University and 3rd place as a best scientist in Saudi Arabia.

Syed M. Eldin is working as an Associate Professor at Center of Research, Faculty of Engineering, University of Malakand, Chakdara, Pakistan.

Hira is working as a Lecturer in the department of biotechnology and microbiology at Bacha Khan University Charsadda, KP, Pakistan.

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Numerical solution of heat and mass transfer using buongionro nanofluid model through a porous stretching sheet impact of variable magnetic,heat source,and temperature conductivity

Abstract

Keywords

Introduction

Mathematical formulation

Numerical procedure

Results and discussion

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Data availability statement

Author biographies

References